Quantitative Finance Exam Solutions
Course Introduction
Quantitative Finance explores the application of mathematical models, statistical techniques, and computational tools to analyze financial markets, value assets and derivatives, manage risk, and construct investment strategies. The course covers foundational topics such as probability theory, stochastic processes, and time series analysis, as well as advanced concepts like option pricing, portfolio optimization, risk management, and algorithmic trading. Students develop the quantitative skills necessary to solve complex financial problems and make data-driven decisions in the finance industry.
Recommended Textbook
Fundamentals of Futures and Options Markets 9th Edition by John C. Hull
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Page 2
Chapter 1: Introduction
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Q1) A one-year call option on a stock with a strike price of $30 costs $3; a one-year put option on the stock with a strike price of $30 costs $4. Suppose that a trader buys two call options and one put option. The breakeven stock price below which the trader makes a profit is
A) $25
B) $28
C) $26
D) $20
Answer: D
Q2) Which of the following is NOT true
A) A call option gives the holder the right to buy an asset by a certain date for a certain price
B) A put option gives the holder the right to sell an asset by a certain date for a certain price
C) The holder of a call or put option must exercise the right to sell or buy an asset
D) The holder of a forward contract is obligated to buy or sell an asset
Answer: C
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Chapter 2: Futures Markets and Central Counterparties
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Q1) A hedger takes a long position in a futures contract on a commodity on November 1, 2012 to hedge an exposure on March 1, 2013. The initial futures price is $60. On December 31, 2012 the futures price is $61. On March 1, 2013 it is $64. The contract is closed out on March 1, 2013. What gain is recognized in the accounting year January 1 to December 31, 2013? Each contract is on 1000 units of the commodity.
A) $0
B) $1,000
C) $3,000
D) $4,000
Answer: D
Q2) Which of the following are cash settled
A) All futures contracts
B) All option contracts
C) Futures on commodities
D) Futures on stock indices
Answer: D
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4
Chapter 3: Hedging Strategies Using Futures
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Q1) Which of the following is necessary for tailing a hedge?
A) Comparing the size in units of the position being hedged with the size in units of the futures contract
B) Comparing the value of the position being hedged with the value of one futures contract
C) Comparing the futures price of the asset being hedged to its forward price
D) None of the above
Answer: B
Q2) Which of the following increases basis risk?
A) A large difference between the futures prices when the hedge is put in place and when it is closed out
B) Dissimilarity between the underlying asset of the futures contract and the hedger's exposure
C) A reduction in the time between the date when the futures contract is closed and its delivery month
D) None of the above
Answer: B
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Chapter 4: Interest Rates
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Q1) An interest rate is 6% per annum with annual compounding. What is the equivalent rate with continuous compounding?
A) 5.79%
B) 6.21%
C) 5.83%
D) 6.18%
Q2) Which of the following is true of LIBOR
A) The LIBOR rate is free of credit risk
B) A LIBOR rate is lower than the Treasury rate when the two have the same maturity
C) It is a rate used when borrowing and lending takes place between banks
D) It is subject to favorable tax treatment in the U.S.
Q3) Which of the following is true?
A) When interest rates in the economy increase, all bond prices increase
B) As its coupon increases, a bond's price decreases
C) Longer maturity bonds are always worth more that shorter maturity bonds when the coupon rates are the same
D) None of the above
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Chapter 5: Determination of Forward and Futures Prices
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Q1) Which of the following is NOT a reason why a short position in a stock is closed out?
A) The investor with the short position chooses to close out the position
B) The lender of the shares issues instructions to close out the position
C) The broker is no longer able to borrow shares from other clients
D) The investor does not maintain margins required on his/her margin account
Q2) Which of the following describes contango?
A) The futures price is below the expected future spot price
B) The futures price is below today's spot price
C) The futures price is a declining function of the time to maturity
D) The futures price is above the expected future spot price
Q3) As inventories of a commodity decline, which of the following is true?
A) The one-year futures price as a percentage of the spot price increases
B) The one-year futures price as a percentage of the spot price decreases
C) The one-year futures price as a percentage of the spot price stays the same
D) Any of the above can happen
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7
Chapter 6: Interest Rate Futures
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Q1) How much is a basis point?
A) 1.0%
B) 0.1%
C) 0.01%
D) 0.001%
Q2) The conversion factor for a bond is approximately
A) The price it would have if all cash flows were discounted at 6% per annum
B) The price it would have if it paid coupons at 6% per annum
C) The price it would have if all cash flows were discounted at 8% per annum
D) The price it would have if it paid coupons at 8% per annum
Q3) Which of following is applicable to corporate bonds in the United States?
A) Actual/360
B) Actual/Actual
C) 30/360
D) Actual/365
Q4) Duration matching immunizes a portfolio against
A) Any parallel shift in the yield curve
B) All shifts in the yield curve
C) Changes in the steepness of the yield curve
D) Small parallel shifts in the yield curve
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Chapter 7: Swaps
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Q1) Which of the following describes the five-year swap rate?
A) The rate on a five-year loan to a AA-rated company
B) The rate on a five-year loan to an A-rated company
C) The rate that can be earned over five years from a series of short-term loans to AA-rated companies
D) The rate that can be earned over five years from a series of short-term loans to A-rated companies
Q2) Since the 2008 credit crisis
A) LIBOR has replaced OIS as the discount rate for non-collateralized swaps
B) OIS has replaced LIBOR as the discount rate, but only for non-collateralized swaps
C) LIBOR has replaced OIS as the discount rate for collateralized swaps
D) OIS has replaced LIBOR as the discount rate for swaps
Q3) Which of the following is usually true
A) OIS rates are less than the corresponding LIBOR rates
B) OIS rates are greater than corresponding LIBOR rates
C) OIS rates are sometimes greater and sometimes less than LIBOR rates
D) OIS rates are equivalent to one-day LIBOR rates
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Page 9
Chapter 8: Securitization and the Credit Crisis of 2007
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Q1) Suppose that ABSs are created from portfolios of subprime mortgages with the following allocation of the principal to tranches: senior 80%, mezzanine 10%, and equity 10%. (The portfolios of subprime mortgages have the same default rates.) An ABS CDO is then created from the mezzanine tranches with the same allocation of principal. Losses on the mortgage portfolio prove to be 16%. What, as a percent of tranche principal, are losses on the senior tranche of the ABS CDO
A) 50%
B) 60%
C) 80%
D) 100%
Q2) Which of the following is NOT true
A) The bonus structure at banks can lead to short-term horizons for decision making
B) Securitization involves the transfer of risk
C) The term "agency costs" describes the situation where the incentives of two parties in a business relationship are not perfectly aligned
D) Correlations decrease in stressed market conditions
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Chapter 9: Mechanics of Options Markets
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Q1) Which of the following describes a short position in an option?
A) A position in an option lasting less than one month
B) A position in an option lasting less than three months
C) A position in an option lasting less than six months
D) A position where an option has been sold
Q2) Which of the following describes LEAPS?
A) Options which are partly American and partly European
B) Options where the strike price changes through time
C) Exchange-traded stock options with longer lives than regular exchange-traded stock options
D) Options on the average stock price during a period of time
Q3) Which of the following is an example of an option class?
A) All calls on a certain stock
B) All calls with a particular strike price on a certain stock
C) All calls with a particular time to maturity on a certain stock
D) All calls with a particular time to maturity and strike price on a certain stock
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11
Chapter 10: Properties of Stock Options
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Q1) Interest rates are zero. A European call with a strike price of $50 and a maturity of one year is worth $6. A European put with a strike price of $50 and a maturity of one year is worth $7. The current stock price is $49. Which of the following is true?
A) The call price is high relative to the put price
B) The put price is high relative to the call price
C) Both the call and put must be mispriced
D) None of the above
Q2) Which of the following is true when dividends are expected?
A) Put-call parity does not hold
B) The basic put-call parity formula can be adjusted by subtracting the present value of expected dividends from the stock price
C) The basic put-call parity formula can be adjusted by adding the present value of expected dividends to the stock price
D) The basic put-call parity formula can be adjusted by subtracting the dividend yield from the interest rate
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Chapter 11: Trading Strategies Involving Options
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Q1) Which of the following is correct?
A) A calendar spread can be created by buying a call and selling a put when the strike prices are the same and the times to maturity are different
B) A calendar spread can be created by buying a put and selling a call when the strike prices are the same and the times to maturity are different
C) A calendar spread can be created by buying a call and selling a call when the strike prices are different and the times to maturity are different
D) A calendar spread can be created by buying a call and selling a call when the strike prices are the same and the times to maturity are different
Q2) When the interest rate is 5% per annum with continuous compounding, which of the following creates a $1000 principal protected note?
A) A one-year zero-coupon bond plus a one-year call option worth about $59
B) A one-year zero-coupon bond plus a one-year call option worth about $49
C) A one-year zero-coupon bond plus a one-year call option worth about $39
D) A one-year zero-coupon bond plus a one-year call option worth about $29
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Chapter 12: Introduction to Binomial Trees
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Q1) Which of the following describes how American options can be valued using a binomial tree?
A) Check whether early exercise is optimal at all nodes where the option is in-the-money
B) Check whether early exercise is optimal at the final nodes
C) Check whether early exercise is optimal at the penultimate nodes and the final nodes
D) None of the above
Q2) A stock is expected to return 10% when the risk-free rate is 4%. What is the correct discount rate to use for the expected payoff on an option in the real world?
A) 4%
B) 10%
C) More than 10%
D) It could be more or less than 10%
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14
Chapter 13: Valuing Stock Options: the Bsm Model
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Q1) A stock price is $100. Volatility is estimated to be 20% per year. What is an estimate of the standard deviation of the change in the stock price in one week?
A) $0.38
B) $2.77
C) $3.02
D) $0.76
Q2) When the non-dividend paying stock price is $20, the strike price is $20, the risk-free rate is 5%, the volatility is 20% and the time to maturity is 3 months which of the following is the price of a European put option on the stock
A) 19.7N(-0.1)-20N(-0.2)
B) 20N(-0.1)-20N(-0.2)
C) 19.7N(-0.2)-20N(-0.1)
D) 20N(-0.2)-20N(-0.1)
Q3) What was the original Black-Scholes-Merton model designed to value?
A) A European option on a stock providing no dividends
B) A European or American option on a stock providing no dividends
C) A European option on any stock
D) A European or American option on any stock
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Chapter 14: Employee Stock Options
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Q1) Which of the following are true of employee stock options?
A) They are commonly valued as though they are regular American options
B) They are commonly valued as though they are regular American options, but with a reduced life.
C) They are commonly valued as though they are regular European option
D) They are commonly valued as though they are regular European options but with a reduced life.
Q2) A company surprises the market with an announcement that it has granted stock options to senior executives. The options are exercised four years later. When does dilution take place?
A) Dilution takes place when the options are exercised
B) Dilution takes place on the announcement date
C) Dilution takes place gradually over the four years
D) There is no dilution
Q3) Which of the following increases the expected life of employee stock options?
A) An increase in the vesting period
B) An increase in employee turnover
C) A fast growth rate for the stock price
D) A tendency for employees to exercise earlier than in the past
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Page 16
Chapter 15: Options on Stock Indices and Currencies
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Q1) For a European call option on a currency, the exchange rate is 1.0000, the strike price is 0.9100, the time to maturity is one year, the domestic risk-free rate is 5% per annum, and the foreign risk-free rate is 3% per annum. How low can the option price be without there being an arbitrage opportunity?
A) 0.1048
B) 0.0900
C) 0.1344
D) 0.1211
Q2) A European at-the-money call option on a currency has four years until maturity. The exchange rate volatility is 10%, the domestic risk-free rate is 2% and the foreign risk-free rate is 5%. The current exchange rate is 1.2000. What is the value of the option?
A) 1.11N(0.7)-0.98N(0.5)
B) 1.11N(-0.7)-0.98N(-0.5)
C) 1.11N(0.7)-0.98N(0.4)
D) 1.11N(-0.06)-0.98N(-0.10)
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Chapter 16: Futures Options and Blacks Model
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Q1) A futures price is currently 40 cents. It is expected to move up to 44 cents or down to 34 cents in the next six months. The risk-free interest rate is 6%. What is the probability of an up movement in a risk-neutral world?
A) 0.4
B) 0.5
C) 0.72
D) 0.6
Q2) Which of the following is true?
A) A futures option is settled daily
B) A futures-style option is settled daily
C) Both a futures option and a futures-style option are settled daily
D) Neither a futures option nor a futures-style option is settled daily
Q3) Which of the following is acquired (in addition to a cash payoff) when the holder of a put futures exercises?
A) A long position in a futures contract
B) A short position in a futures contract
C) A long position in the underlying asset
D) A short position in the underlying asset
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18
Chapter 17: The Greek Letters
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Q1) Which of the following is NOT a letter in the Greek alphabet?
A) delta
B) rho
C) vega
D) gamma
Q2) Which of the following is NOT true about gamma?
A) A highly positive or highly negative value of gamma indicates that a portfolio needs frequent rebalancing to stay delta neutral
B) The magnitude of gamma is a measure of the curvature of the portfolio value as a function of the underlying asset price
C) A big positive value for gamma indicates that a big movement in the asset price in either direction will lead to a loss
D) A long position in either a call or a put has a positive gamma
Q3) Vega tends to be high for which of the following
A) At-the money options
B) Out-of-the money options
C) In-the-money options
D) Options with a short time to maturity
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Chapter 18: Binomial Trees in Practice
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Q1) When the stock price is 20 and the present value of dividends is 2, which of the following is the recommended way of constructing a tree?
A) Draw a tree for an initial stock price of 20 and subtract the present value of future dividends at each node
B) Draw a tree for an initial stock price of 22 and subtract the present value of future dividends at each node
C) Draw a tree with an initial stock price of 18 and add the present value of future dividends at each node
D) Draw a tree with an initial stock price of 18 and add 2 at each node
Q2) What is the recommended way of making interest rates a function of time in a Cox, Ross, Rubinstein tree?
A) Make u a function of time
B) Make p a function of time
C) Make u and p a function of time
D) Make the lengths of the time steps unequal
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20
Chapter 19: Volatility Smiles
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Q1) Which of the following is true when the tails of a future foreign currency distribution are compared with those of a lognormal distribution with the same mean and standard deviation?
A) The left tail and right tail are thinner
B) The left tail is thinner and the right tail is fatter
C) The right tail is thinner and the left tail is fatter
D) Both tails are fatter
Q2) Which of the following causes a volatility smile that is a ``frown``?
A) There is a small probability of a large stock price decrease in one week
B) There is a small probability of a large stock price increase in one week
C) The outcome of a lawsuit (roughly equal chance of being favorable or unfavorable) will create a large movement up or down in one week
D) None of the above
Q3) Which of the following is true as time to maturity increases?
A) The volatility smile for currency options tends to become more pronounced
B) The volatility smile for currency options tends to become less pronounced
C) The volatility smile for currency options first becomes less pronounced and then becomes more pronounced
D) The volatility smile for currency options remains approximately the same
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Page 21
Chapter 20: Value at Risk and Expected Shortfall
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Q1) At the end of Thursday, the estimated covariance between assets A and B is 0.0001. During Friday asset A produces a return of 3% and asset B produces a return of zero. An EWMA model with lambda equal to 0.9 is used. What is an estimate of the covariance at the end of Friday?
A) 0.000090
B) 0.000081
C) 0.000100
D) 0.000095
Q2) Which of the following is true?
A) The quadratic model approximates daily changes in using delta and gamma
B) The quadratic model approximates daily changes using delta, but not gamma
C) The quadratic model approximates daily changes using gamma, but not delta
D) None of the above
Q3) If the volatility for a portfolio is 20% per year, what is the volatility per quarter?
A) 20%
B) 10%
C) 5%
D) 2%
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22
Chapter 21: Interest Rate Options
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Q1) Which of the following is true?
A) A callable bond allows the lender to ask for the principal to be repaid early
B) A callable bond allows the borrower to repay the principal early
C) A callable bond is a bond with an embedded stock option
D) None of the above
Q2) Which of the following is assumed to be lognormal when a bond option is valued?
A) A future bond price
B) A future swap rate
C) A future short-term rate
D) A future bond yield
Q3) A five-year cap is reset annually period. The cap rate is 3% and the notional principal is $100 million. The 12-month LIBOR interest rate for the third year proves to be 5%. Which of the following is approximately true?
A) The resulting payoff is $2 million at the beginning of the third year
B) The resulting payoff is $2 million at the end of the fifth year
C) The resulting payoff is $2 million at the end of the third year
D) The resulting payoff is $2 million half way through the third year
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Page 23
Chapter 22: Exotic Options and Other Nonstandard Products
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Q1) When can Bermudan options be exercised?
A) Any time during the life of the options
B) Any time after a certain date up to the end of the life of the life
C) Any time before a certain date or at the end of the option's life
D) On dates specified at the start of the option
Q2) As the barrier is observed more frequently, a knock out option becomes which of the following
A) More valuable
B) Less valuable
C) There is no effect on value
D) May become more valuable or less valuable
Q3) An employer has promised that it will grant employees three year options in one year's time and that the options will be at the money at the time they are granted. What describes these options?
A) Chooser options
B) Forward start options
C) Compound options
D) Shout options
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Chapter 23: Credit Derivatives
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Q1) A CDS with a number of reference entities provides a payoff when any of the reference entities defaults. What is a name for this CDS?
A) Binary CDS
B) Add-up Basket CDS
C) First-to-Default CDS
D) n-to-Default CDS
Q2) Which of the following happens when the default correlation of the companies underlying a CDO increases?
A) The value of the senior tranche and the equity tranche to the protection buyer both increase
B) The value of the senior tranche and the equity tranche to the protection buyer both decrease
C) The value of the senior tranche to the protection buyer decreases and the value of the equity tranche to the protection buyer increases
D) The value of the senior tranche to the protection buyer increases and the value of the equity tranche to the protection buyer decreases
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Chapter 24: Weather, Energy, and Insurance Derivatives
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Q1) Which of the following describes the period during which a "5 times 8" contract provides electricity?
A) From 11pm to 7am on five successive days
B) From 8am to 4pm on five successive days
C) For any 5 hours of a day on 8 successive days
D) For any 8 hours of a day in five successive days
Q2) Which of the following is NOT seasonal?
A) Spot electricity
B) Spot natural gas
C) Electricity futures prices
D) Spot price of corn
Q3) Which of the following might we expect to be the result of global warming?
A) An decrease in observed CDDs
B) An increase in observed CDDs
C) An increase in observed HDDs
D) None of the above
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